Lemma 66.18.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $y \in |Y|$ and assume that $y$ is represented by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to Y$. Then $|X_ k| \to |X|$ is a homeomorphism onto $f^{-1}(\{ y\} ) \subset |X|$ with induced topology.

** Fibers of field points of algebraic spaces have the expected Zariski topologies. **

**Proof.**
We will use Properties of Spaces, Lemma 64.16.7 and Morphisms of Spaces, Lemma 65.10.9 without further mention. Let $V \to Y$ be an étale morphism with $V$ affine such that there exists a $v \in V$ mapping to $y$. Since $\mathop{\mathrm{Spec}}(k) \to Y$ is quasi-compact there are a finite number of points of $V$ mapping to $y$ (Lemma 66.4.5). After shrinking $V$ we may assume $v$ is the only one. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Consider the commutative diagram

Since $U_ v \to U_ V$ identifies $U_ v$ with a subset of $U_ V$ with the induced topology (Schemes, Lemma 26.18.5), and since $|U_ V| \to |X_ V|$ and $|U_ v| \to |X_ v|$ are surjective and open, we see that $|X_ v| \to |X_ V|$ is a homeomorphism onto its image (with induced topology). On the other hand, the inverse image of $f^{-1}(\{ y\} )$ under the open map $|X_ V| \to |X|$ is equal to $|X_ v|$. We conclude that $|X_ v| \to f^{-1}(\{ y\} )$ is open. The morphism $X_ v \to X$ factors through $X_ k$ and $|X_ k| \to |X|$ is injective with image $f^{-1}(\{ y\} )$ by Properties of Spaces, Lemma 64.4.3. Using $|X_ v| \to |X_ k| \to f^{-1}(\{ y\} )$ the lemma follows because $X_ v \to X_ k$ is surjective. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (4)

Comment #1110 by Evan Warner on

Comment #1111 by Evan Warner on

Comment #1112 by Evan Warner on

Comment #1138 by Johan on